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MNRAS 453, 1288–1296 (2015) doi:10.1093/mnras/stv1725

Asteroid 2015 DB216: a recurring co-orbital companion to

C. de la Fuente Marcos‹ and R. de la Fuente Marcos Apartado de Correos 3413, E-28080 Madrid, Spain

Accepted 2015 July 27. Received 2015 July 7; in original form 2015 June 3

ABSTRACT Minor bodies trapped in 1:1 co-orbital with a host could be relevant to explain

the origin of captured . Among the giant , Uranus has one of the smallest known Downloaded from populations of co-orbitals, three objects, and all of them are short-lived. 2015 DB216 has an that matches well that of Uranus, and here we investigate its dynamical state. Direct N-body calculations are used to assess the current status of this object, reconstruct its immediate dynamical past, and explore its future orbital evolution. A covariance matrix- based Monte Carlo scheme is presented and applied to study its short-term stability. We find http://mnras.oxfordjournals.org/ that 2015 DB216 is trapped in a temporary co-orbital with Uranus, the fourth known minor body to do so. A detailed analysis of its dynamical evolution shows that it is an unstable but recurring co-orbital companion to Uranus. It currently follows an asymmetric horseshoe trajectory that will last for at least 10 kyr, but it may remain inside Uranus’ co-orbital zone for millions of years. As in the case of other transient Uranian co-orbitals, complex multibody ephemeral resonances trigger the switching between the various resonant co- orbital states. The new Uranian co-orbital exhibits a secular behaviour markedly different from ◦ that of the other known Uranian co-orbitals because of its higher inclination, nearly 38 .Given at Northwestern University Library on February 2, 2016 its rather unusual discovery circumstances, the presence of 2015 DB216 hints at the existence of a relatively large population of objects moving in similar . Key words: methods: numerical – methods: statistical – – minor planets, : individual: 2015 DB216 – planets and satellites: individual: Uranus.

et al. 2013; de la Fuente Marcos & de la Fuente Marcos 2014). Due 1 INTRODUCTION to its present short data-arc (85 d), 2010 EU65 is better described For over a century, co-orbitals – or minor bodies trapped in a 1:1 as a candidate. Asteroids Crantor and 2010 EU65 follow horse- mean motion resonance with a host planet – have been regarded as shoe orbits, and 2011 QF99 is an L4 . Consistently, Uranus mere interesting dynamical curiosities (Jackson 1913; Henon 1969; has also a small population of irregular satellites, significantly Namouni 1999). This view is now changing considerably; in fact, smaller than that of or (Grav et al. 2003; Sheppard, the heliocentric 1:1 co- could be an efficient mech- Jewitt & Kleyna 2005). Most Uranian irregular satellites are ret- anism for capture of satellites by a planet and therefore explain the rograde and their large spread in semimajor axis suggest that they origin of some irregular (Kortenkamp 2005). This theoretical formed independently (Nesvornyetal.´ 2003); orbits with inclina- possibility – that of being a feasible dynamical pathway to capture tions in the range (80◦, 100◦) are unstable due to the Kozai resonance satellites, at least temporarily – was dramatically vindicated when (Kozai 1962). For the particular case of Uranus’ Trojans, Dvorak, a co-orbital of the , 2006 RH120, remained as natural Bazso´ & Zhou (2010) have also found that the stability depends of our planet for about a year starting in 2006 June (Kwiatkowski on the and only the inclination intervals (0◦,7◦), et al. 2009; Granvik, Vaubaillon & Jedicke 2012). (9◦,13◦), (31◦,36◦), and (38◦,50◦) seem to be stable. Asteroid Uranus has one of the smallest known populations of co-orbitals 2011 QF99 appears to inhabit one of these stable islands at an incli- and all of them are relatively short-lived (de la Fuente Marcos & de nation of nearly 11◦ (de la Fuente Marcos & de la Fuente Marcos la Fuente Marcos 2014). So far, Uranus had only three known co- 2014). The stability of Uranian Trojans had been previously studied orbitals: (2002 GO9) (Gallardo 2006; de la Fuente by Marzari, Tricarico & Scholl (2003), and by Nesvorny´ & Dones Marcos & de la Fuente Marcos 2013), 2010 EU65 (de la Fuente Mar- (2002) and Holman & Wisdom (1993) before them. cos & de la Fuente Marcos 2013), and 2011 QF99 (Alexandersen Here, we present a recently discovered object, 2015 DB216,that is also trapped in a 1:1 mean motion resonance with Uranus. This minor body exhibits some dynamical features that separate it from E-mail: [email protected] the previously known Uranian co-orbitals. This paper is organized

C 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society A recurring co-orbital to Uranus 1289

Table 1. Heliocentric Keplerian of 2015 DB216 used elements provided by the JPL On-line Data Service3 in this research. The is based on 28 observations spanning (Giorgini et al. 1996) and initial positions and velocities based on a data-arc of 4200 d or 11.50 yr, from 2003 October 21 to 2015 the DE405 planetary orbital ephemerides (Standish 1998) referred April 23. Values include the 1σ uncertainty. The orbit is computed to the barycentre of the Solar system. Besides the orbital calcula- at JD 245 7000.5 that corresponds to 0:00 UT on 2014 December 9 (J2000.0 and equinox) and it is t = 0 in the figures. Source: tions completed using the nominal elements in Table 1,wehave JPL Small-Body Database. performed 50 control simulations with sets of orbital elements ob- tained from the nominal ones as described in the following section, Semimajor axis, a (au) = 19.204 ± 0.005 all of them for 0.5 Myr forward and backwards in time. Two more Eccentricity, e = 0.323 95 ± 0.000 13 sets of 100 control orbits each have been integrated for just 5 kyr into Inclination, i (◦) = 37.7173 ± 0.0003 the past and the future to better characterize its short-term stability. Longitude of the ascending node,  (◦) = 6.2679 ± 0.0003 Argument of perihelion, ω (◦) = 237.75 ± 0.03 , M (◦) = 302.52 ± 0.04 3 GENERATING CONTROL ORBITS Perihelion, q (au) = 12.9832 ± 0.0013 WITH MONTE CARLO AND THE Aphelion, Q (au) = 25.426 ± 0.007 COVARIANCE MATRIX Absolute , H (mag) = 8.3 ± 0.4 Downloaded from For a given minor body, the orbital elements are a coordinate in six- dimensional space (assuming as we do that non-gravitational forces can be neglected), which represents the location where samples of as follows. In Section 2, we briefly discuss both the data and the control orbits are most likely to be generated. This is analogous to the peak of the Gaussian curve for a typical one-dimensional or

numerical model used in our calculations. The topic of generating http://mnras.oxfordjournals.org/ control orbits compatible with the available observations and its univariate normal distribution. The multivariate normal distribution implications is considered in Section 3. The current status of 2015 is a generalization of the one-dimensional normal distribution to higher dimensions. Instead of being specified by its mean value and DB216 is studied in Section 4, where its dynamical past and future orbital evolution are also investigated. Section 5 discusses our re- variance, such a distribution is characterized by its mean (a vector sults and their possible significance. The stability of the co-orbital with the mean of the multidimensional distribution) and covariance matrix, which defines a hyperellipsoid in multidimensional space. realm located in the neighbourhood of 2015 DB216 is tentatively explored in Section 6. A summary of our conclusions is given in The values of the elements of the covariance matrix indicate the Section 7. level to which two given variables vary together. For a particular object, both mean and covariance matrix are computed from the

observations. at Northwestern University Library on February 2, 2016 2 DATA AND METHODOLOGY When studying the stability of the orbital solution of a certain , we can compute the orbital elements of the con- Asteroid 2015 DB216 was discovered on 2015 February 27 at Mt trol orbits varying them randomly, within the ranges defined by = Lemmon Survey. With a value of the semimajor axis a 19.20 au, their mean values and standard deviations. For example, a new = this Centaur moves in an eccentric, e 0.32, and highly inclined value of the can be found using the expression path, i = 37◦.72. With such an orbit, close encounters are only pos- ec = e + σ e ri,whereec is the eccentricity of the control orbit, e is sible with Uranus as its perihelion is well beyond Saturn’s aphelion the mean value of the eccentricity (nominal orbit), σ e is the standard and its aphelion far from ’s perihelion. It is a relatively deviation of e (nominal orbit), and r is a (pseudo) random number = i large object with H 8.3 mag which translates into a diameter in with normal distribution in the range −1 to 1. In statistical terms, the range 46–145 km for an assumed of 0.40–0.04. Its period the univariate Gaussian distribution results from adding a standard of revolution around the , approximately 84.16 yr at present, is Gaussian variate with mean zero and variance one multiplied by the very close to that of Uranus which is suggestive of an object that standard deviation, to the mean value. This is equivalent to consid- moves co-orbital with the . Its current orbit is statisti- ering a number of different virtual minor planets moving in similar cally robust because six images acquired by the Sloan orbits, not a sample of control orbits incarnated from a set of ob- Digital Sky Survey (SDSS) at Apache Point late in 2003 have been servations obtained for a single minor planet. If the control orbits found. The heliocentric Keplerian osculating orbital elements and are meant to be compatible with actual observations, we have to uncertainties in Table 1 are based on 28 observations for a data-arc consider how the elements affect each other using the covariance span of 4200 d and they have been obtained from the Jet Propulsion matrix or e.g. following the procedure described in Sitarski (1998, 1 Laboratory (JPL) Small-Body Database. 1999, 2006). In order to assess the dynamical status of 2015 DB216,weuse The methodology used in this paper is an implementation of the Hermite integration scheme described by Makino (1991)and the classical Monte Carlo using the Covariance Matrix (MCCM; implemented by Aarseth (2003). The standard version of this direct Bordovitsyna, Avdyushev & Chernitsov 2001; Avdyushev & 2 N-body code is publicly available from the IoA website. Our phys- Banschikova 2007) approach, i.e. a Monte Carlo process creates ical model includes the perturbations by the eight major planets, control orbits with initial parameters from the nominal orbit adding the , the barycentre of the system, and the three random noise on each initial orbital element making use of the co- largest asteroids; additional details can be found in de la Fuente variance matrix. The MCCM approach considers that the estimated Marcos & de la Fuente Marcos (2012). To compute accurate initial parameters are Gaussian random variables with mean values those positions and velocities we used the heliocentric ecliptic Keplerian of the nominal orbit and covariance matrix obtained via the least- squares method applied to the available observations. Assuming a

1 http://ssd.jpl.nasa.gov/sbdb.cgi 2 http://www.ast.cam.ac.uk/∼sverre/web/pages/nbody.htm 3 http://ssd.jpl.nasa.gov/?planet_pos

MNRAS 453, 1288–1296 (2015) 1290 C. de la Fuente Marcos and R. de la Fuente Marcos covariance matrix as computed by the JPL Solar System Dynamics Group, Horizons On-Line System, the vector including the mean values of the orbital parameters at a given epoch t0 is of the form v = (e,q,τ,,ω,i); the perihelion is given by the expres- sion q = a (1 − e). If C is the covariance matrix at the same epoch associated with the nominal orbital solution that is symmetric and positive-semidefinite, then C = AAT,whereA is a lower triangular matrix with real and positive diagonal elements, AT is the transpose of A. In the particular case studied here, these matrices are 6 × 6. If the elements of C are written as cij and those of A as aij,where those are the entries in the ith row and jth column, they are related by the following expressions: √ a11 = c11 a21 = c12/a11 Downloaded from  ω a31 = c13/a11 Figure 1. Time evolution of the orbital elements a, e, i, ,and of 2015 DB . The thick black curve shows the average evolution of 100 control a = c /a 216 41 14 11 orbits, the thin red curves display the ranges in the values of the parameters at the given time. Control orbits depicted in the left-hand panels have been a51 = c15/a11

computed as described in the text, using the covariance matrix (equations 2). http://mnras.oxfordjournals.org/ a61 = c16/a11 Those displayed in the right-hand panels have been computed without taking  into account the covariance matrix (equations 3). a = c − a2 22 22 21 In contrast, the equivalent classical – but statistically wrong – ex- a32 = (c23 − a21 a31)/a22 pressions commonly used to generate control orbits are given by a42 = (c24 − a21 a41)/a22 ec = e + σe r1 a52 = (c25 − a21 a51)/a22 qc = q + σq r2 a62 = (c26 − a21 a61)/a22 at Northwestern University Library on February 2, 2016  τc = τ + στ r3 2 2 a33 = c33 − a − a 31 32 c =  + σ r4 a43 = (c34 − a31 a41 − a32 a42)/a33 ωc = ω + σω r5 a53 = (c35 − a31 a51 − a32 a52)/a33 ic = i + σi r6 . (3) a63 = (c36 − a31 a61 − a32 a62)/a33  A comparison between the results of the evolution of a sample of a = c − a2 − a2 − a2 control orbits generated using equations (2) and (3) for the particular 44 44 41 42 43 case of 2015 DB216 appears in Fig. 1, left-hand and right-hand a54 = (c45 − a41 a51 − a42 a52 − a43 a53)/a44 panels, respectively. In our calculations, the Box–Muller method (Press et al. 2007) was used to generate random numbers with a a64 = (c46 − a41 a61 − a42 a62 − a43 a63)/a44  normal distribution. It is obvious that, at least for this particular a = c − a2 − a2 − a2 − a2 object, the difference is not very significant. However and for very 55 55 51 52 53 54 precise orbits, the outcomes from these two approaches could be a65 = (c56 − a51 a61 − a52 a62 − a53 a63 − a54 a64)/a55 very different (see e.g. fig. 5 in Sitarski 1998); creating control  orbits by randomly varying the nominal orbital elements in range a = c − a2 − a2 − a2 − a2 − a2 . 66 66 61 62 63 64 65 (1) of their mean errors (equations 3) is not recommended in that case.

If r is a vector made of univariate Gaussian random numbers (com- = ponents ri with i 1, 6), the required multivariate Gaussian random 4 ASTEROID 2015 DB216: samples – i.e. the sets of initial orbital elements of the control orbits DYNAMICAL EVOLUTION – are given by the expressions (assuming the structure provided by In order to assess the dynamical status of 2015 DB216, we the JPL Horizons On-Line Ephemeris System), vc = v + A r: on the study of the librational behaviour of the relative mean lon- gitude λ = λ − λ ,whereλ and λ are the mean longitudes of ec = e + a11 r1 r U U the object and Uranus, respectively; λ = M +  + ω,whereM q = q + a r + a r c 22 2 21 1 is the mean anomaly,  is the longitude of the ascending node, ◦ and ω is the argument of perihelion. If λr oscillates around 0 ,the τc = τ + a33 r3 + a32 r2 + a31 r1 object is considered a quasi-satellite; Trojan bodies are character-  =  + a r + a r + a r + a r ◦ ◦ ◦ c 44 4 43 3 42 2 41 1 ized by λr librating around +60 (L4 Trojan) or −60 (or 300 , λ ω = ω + a r + a r + a r + a r + a r L5 Trojan); finally, an object whose r oscillates with amplitude c 55 5 54 4 53 3 52 2 51 1 >180◦ follows a (see e.g. Murray & Dermott 1999). ic = i + a66 r6 + a65 r5 + a64 r4 + a63 r3 + a62 r2 + a61 r1 . (2) Quasi-satellites are not true gravitationally bound satellites but

MNRAS 453, 1288–1296 (2015) A recurring co-orbital to Uranus 1291

Marcos & de la Fuente Marcos (2014). As other Uranian co-orbitals do, 2015 DB216 moves in near resonance with the other three giant planets: 1:7 with Jupiter, 7:20 with Saturn, and 2:1 with Neptune. Fig. 4 shows the behaviour of the resonant arguments σ J = 7λ − λJ − 6 , σ S = 20λ − 7λS − 13 ,andσ N = λ − 2λN +  , where λJ is the of Jupiter, λS is the mean longitude of Saturn, λN is the mean longitude of Neptune, and  =  + ω is the longitude of the perihelion of 2015 DB216. The plot (similar to figs 6 and 7 in de la Fuente Marcos & de la Fuente Marcos 2014) clearly indicates that transitions are quickly triggered when multiple mean motion resonances work in unison. In Fig. 4,an Figure 2. The motion of 2015 DB216 during the time interval (0, 10) kyr originally passing orbit becomes a horseshoe path after σ J and σ N projected on to the ecliptic plane in a coordinate system rotating with Uranus stop circulating; prior to the ejection from the horseshoe-like path, (red curve, left-hand panel). The orbit and the position of Uranus are also the same scenario is observed. The dynamical role of three-body plotted (blue curve). In this frame of reference, and as a result of its non- mean motion resonances has been recently explored by Gallardo negligible eccentricity, Uranus describes a small ellipse (black curve). The Downloaded from (2014). Marzari et al. (2003) already pointed out that three-body associated values of the resonant angle, λr, are also displayed (right-hand panel). resonances could be a source of instability for Uranian co-orbitals, in particular Trojans. It may be argued that it is unclear from Fig. 4 that overlap- appear to orbit the host planet like a retrograde satellite. If λr can ping mean motion resonances are responsible for the transitions take any value (circulates), we speak of passing orbits. between co-orbital states or the activation/deactivation of the ob- http://mnras.oxfordjournals.org/ Our N-body integrations show that 2015 DB216 is currently a co- served librational dynamics. On strictly theoretical grounds, this is orbital companion to Uranus and moves in an asymmetric horseshoe to be expected as the orbital architecture of the giant planets is not orbit with a period of about 11 kyr (see Fig. 2, right-hand panel); in random. Ito & Tanikawa (2002) and Tanikawa & Ito (2007)have this case, asymmetric means that the resonant angle, λr, goes beyond pointed out that Jupiter affects the motions of Uranus and Neptune 0◦ reaching an offset of around −40◦ at nearly 9 kyr. The without the connection of Saturn and that secular perturbations may left-hand panel in Fig. 2 depicts the trajectory of 2015 DB216 viewed be nullified in such context. To explore this issue further, we have in a frame of reference corotating with Uranus. Fig. 3 displays the recomputed the short-term orbital evolution of the nominal orbit dynamical evolution of various parameters for three representative of 2015 DB216 using increasingly complex physical models. Inte-

orbits: the nominal one (central panels) and those of two additional grating the three-body problem – Sun, Uranus, and 2015 DB216 at Northwestern University Library on February 2, 2016 orbits which are most different from the previous one, and have been – we observe asymmetric horseshoe evolution with no transitions. obtained adding (+) or subtracting (−) six times the uncertainty Computing the evolution of the four-body problem – Sun, Jupiter, from the orbital parameters (the six elements) in Table 1. All the Uranus, and 2015 DB216 – a transition from asymmetric horseshoe control orbits show consistent behaviour within a few thousand to L4 Trojan at about 15 kyr is recorded. The alternative four-body years of t = 0 (see Fig. 1). Its e-folding time, or characteristic time- problem – Sun, Saturn, Uranus, and 2015 DB216 – results in an scale on which two arbitrarily close orbits diverge exponentially, is L4 Trojan path with no transitions. A similar result is observed a few thousand years both in the past and the future. The evolution for the case Sun, Uranus, Neptune, and 2015 DB216. The six-body of the control orbits exhibits very similar behaviour of all the orbital problem – Sun, Jupiter, Saturn, Uranus, Neptune, and 2015 DB216 elements within the time frame (−3, 3) kyr (see Fig. 1). – results in an asymmetric horseshoe transitioning to a passing Asteroid 2015 DB216 currently occupies (see Fig. 3, E panels) orbit, but somewhat earlier than observed in Fig. 4. It appears a band of instability between the two stable islands in inclination, obvious that in order to turn the asymmetric horseshoe libration (31◦,36◦)and(38◦,50◦), described in Dvorak et al. (2010) for Ura- into a passing orbit, superposition of mean motion resonances is nian Trojans. However, the figure shows that the inclination of this required. asteroid is high enough to avoid close encounters with Uranus when A more systematic exploration of the various five-, six-, and the relative mean longitude approaches zero, i.e. close encounters higher-multiplicity-body problems shows that the details of the tran- with Uranus (or any other body) are not responsible for the acti- sitions are strongly dependent on the number of distant perturbers vation and deactivation of the co-orbital behaviour of this object. included in the simulations. The dynamical evolution of these ob- Very few close encounters with Uranus have been observed during jects is unusually sensitive to the physical model used to perform the simulated time (examples appear in the A-left-hand and centre the calculations. Removing the three asteroids and the Pluto–Charon panels of Fig. 3). However, multiple and repetitive short co-orbital system does not have a major observable impact on the outcome episodes of the Trojan, quasi-satellite and horseshoe type are ob- of the simulations both in terms of the timing and the types of the served in Fig. 3. Recurrent co-orbital episodes in which the relative observed transitions (the evolution displayed in Fig. 4 remains very mean longitude librates for several cycles and then circulates for a nearly the same). However, stripping planets from the model – even few more cycles before restarting libration once again, are the sign- or – has immediate effects on the orbital evolution of post of a type of dynamical behaviour known as resonance angle these recurring Uranian co-orbitals. For example, removing Mer- nodding, see e.g. Ketchum, Adams & Bloch (2013); nodding often cury from the calculations triggers a transition from asymmetric occurs when a small body is in an external (near) mean motion horseshoe to L4 Trojan at about 15 kyr and back to asymmetric resonance with a larger planet. In our case, the situation is more horseshoe a few kyr later. This analysis indicates that any numeri- complicated because we have multiple distant perturbers. cal study of these objects that is not using the full set of planets may Transitions in and out or between the various co-orbital states arrive to unrealistic conclusions regarding the stability and dynam- are not triggered by encounters but result from complex multibody ical evolution of these objects. This is consistent with the analysis ephemeral mean motion resonances as described in de la Fuente in Tanikawa & Ito (2007). Published works like those of Marzari

MNRAS 453, 1288–1296 (2015) 1292 C. de la Fuente Marcos and R. de la Fuente Marcos Downloaded from http://mnras.oxfordjournals.org/ at Northwestern University Library on February 2, 2016

Figure 3. Comparative dynamical evolution of various parameters for the nominal orbit of 2015 DB216 as presented in Table 1 (central panels) and two representative orbits that are most different from the nominal one (see the text for details). The distance from Uranus (A panels); the value of the radius of Uranus, 0.4469 au, is displayed as a red line. The resonant angle, λr (B panels). The orbital elements a (C panels, the value of the semimajor axis of Uranus appears as a red line), e (D panels), i (E panels), and ω (F panels). The distances to the descending (thick line) and ascending nodes (dotted line) appear in the G panels. Saturn and Neptune aphelion and perihelion distances are also shown as red lines. et al. (2003) or Alexandersen et al. (2013) made use of a five-body the longitude of the perihelion of 2015 DB216,  =  + ω, is only model including the Sun and the four outer planets. in secular resonance with Neptune and for a limited time. The value ◦ As for the secular behaviour (see Fig. 5), it is markedly different of  =  −  N librates around 180 . The absence of apsidal from the one described for other Uranian co-orbitals in de la Fuente corotation resonances (see Lee & Peale 2002; Beauge,´ Ferraz-Mello Marcos & de la Fuente Marcos (2014). The precession frequency of & Michtchenko 2003) probably translates into increased stability

MNRAS 453, 1288–1296 (2015) A recurring co-orbital to Uranus 1293

Table 2. Equatorial coordinates and apparent magnitudes (with filter) at discovery time of known Uranian co-orbitals and candidates (J2000.0 ecliptic and equinox). Source: MPC Database.

Object α (h:m:s) δ (◦::) m (mag)

1999 HD12 12:31:54.80 −01:03:07.9 22.9 (R) (83982) Crantor 14:10:43.80 +01:24:45.5 19.2 (R) 2002 VG131 00:54:57.98 +12:07:52.4 22.5 (R) 2010 EU65 12:15:58.608 −02:07:16.66 21.2 (R) 2011 QF99 01:57:34.729 +14:35:44.64 22.8 (r)

2015 DB216 (SDSS, 2003) 08:29:42.21 +57:19:08.2 22.4 (V) 2015 DB216 (MLS, 2015) 11:09:56.70 +29:31:01.6 20.5 (V)

of this object when compared with other Uranian co-orbitals. Even

if evidently chaotic, its dynamical evolution appears to be relatively Downloaded from stable and the object may remain in the neighbourhood of Uranus co-orbital region for millions of years. On the other hand, during the quasi-satellite episode observed in Fig. 3, right-hand panels, the object exhibits Kozai-like dynamics with ω librating around 270◦ for about 100 kyr. http://mnras.oxfordjournals.org/

5 DISCUSSION Our analysis suggests that, even if submitted to chaotic dynamics, Figure 4. Resonant arguments σ = 7λ − λ − 6 (top panel), σ = 20λ − J J S this object may be intrinsically more stable than any of the previ- 7λS − 13 (middle panel), and σ N = λ − 2λN +  (bottom panel) plotted against time for the time interval (−10, 30) kyr. The relative mean longitude ously known Uranian co-orbitals, but there is an additional piece with respect to Uranus appears as a thick blue line. The angle σ S alternates of robust evidence in favour of this interpretation. Asteroid 2015 between circulation and asymmetric libration, indicating that the motion is DB216 was serendipitously discovered by a survey aimed at finding chaotic. The observed resonant evolution is consistent across control orbits. near-Earth objects (NEOs), the Mt Lemmon Survey (MLS), that is part of the (CSS)4 and precovered from obser- at Northwestern University Library on February 2, 2016 vations acquired by the SDSS,5 a project aimed at creating the most detailed three-dimensional maps of the ever made after imaging about one-third of the sky. Therefore, its observation (past and present) was not the result of careful planning like it was the case of the discovery of 2011 QF99 (Alexandersen et al. 2013). In de la Fuente Marcos & de la Fuente Marcos (2014),section8,we studied the discovery circumstances of known Uranian co-orbitals and candidates. All of them have been found at declinations in the ◦ ◦ range −2 to +15 (see Table 2). In sharp contrast, 2015 DB216 was observed at declination +57◦.3 by SDSS in 2003 and at +29◦.5by MLS in 2015. In de la Fuente Marcos & de la Fuente Marcos (2014), fig. 18, an observational bias regarding the observation of Uranian co-orbitals was pointed out, that co-orbitals reaching perigee (or perihelion) near declination 0◦ are nearly six times more likely to be found than those reaching perigee at declinations ± 60◦,ifthey do exist. Table 2 includes data from the (MPC) Database6 and clearly shows that, among Uranian co-orbitals, 2015 DB216 is a puzzling outlier. Assuming that this object is not a sta- tistical accident, its presence hints at the existence of a significant population of objects moving in similar orbits, perhaps an order of magnitude larger than current models predict for regular Uranian co-orbitals. The absence of secular perturbations by Jupiter and Uranus found for 2015 DB216 may probably explain the relative stability of this putative population. It could be the case that – after all – Uranus may host a large population of (transient but recurring) Figure 5. Time evolution of the relative longitude of the perihelion, co-orbitals, but their orbits may be characterized by high orbital  , of 2015 DB216 with respect to the giant planets: referred to Jupiter ( −  J), to Saturn ( −  S), to Uranus ( −  U), and to Neptune 4 ( −  N). The relative longitudes circulate over the entire simulated pe- http://www.lpl.arizona.edu/css/css_facilities.html riod with the exception of that of Neptune. These results are for the nominal 5 http://www.sdss.org orbit in Table 1. 6 http://www.minorplanetcenter.net/db_search

MNRAS 453, 1288–1296 (2015) 1294 C. de la Fuente Marcos and R. de la Fuente Marcos Downloaded from

Figure 6. Mean value (bottom panel) and standard deviation (top panel) of Figure 8. Same as Fig. 6 but for the initial value of the inclination. the resonant angle, λr, as a function of the initial value of the semimajor axis. The value of the standard deviation for a continuous uniform distribution of ◦ ◦ maximum value 180◦ and minimum value −180◦ is also indicated, ∼104◦. and ω are chosen in the range 0 –360 . An object following a strict passing orbit has an average value of the resonant angle close to 0◦

and its associated standard deviation is nearly 104◦ (also displayed http://mnras.oxfordjournals.org/ on the top panel of the figures as a dashed line); the value of the variance of a continuous uniform distribution of maximum value xmax and minimum value xmin is given by the expression (xmax − 2 xmin) /12, and the mean value is (xmax + xmin)/2. Consistently with our analysis of the dynamics of 2015 DB216 in which recurring co- orbital episodes of various types are observed, the fictitious orbits studied here show values of the standard deviation of the resonant angle in obvious conflict with those expected in a non-librational

scenario. Therefore, there is a robust theoretical ground to assume at Northwestern University Library on February 2, 2016 that such population of high orbital inclination Uranian recurring co-orbitals may exist. Fig. 8 shows that the initial value of the orbital inclination does λ  σ not have a major impact on r or λr (for the ranges of the values of the orbital elements considered here), but Fig. 7 indicates that the initial value of the orbital eccentricity has a major influence on Figure 7. Same as Fig. 6 but for the initial value of the eccentricity. the subsequent evolution of the test orbit. For values in the range 0.1–0.3 the magnitude of the standard deviation of the resonant inclinations. The discovery of 2015 DB216 parallels that of 83982 angle tends to be significantly lower when recurring co-orbital be- Crantor (2002 GO9), found by the Near-Earth Asteroid Tracking haviour appears. These orbits are inherently more stable as their (NEAT) project at in 2002 and precovered perihelia and aphelia are less directly perturbed. They are associ- from images obtained in 2001 by the Air Force Maui Optical and ated with Trojan and quasi-satellite co-orbital states. Supercomputing (AMOS) observatory and SDSS. Returning to the issue of the actual extension of the Uranian co-orbital zone for these high-inclination, transient but recurring co-orbitals, the distribution in semimajor axis (the initial value) 6 EXPLORING THE ORBITAL DOMAIN NEAR ◦ ◦ for test orbits with σλ ∈ (100 , 108 ) (top panel) and outside that 2015 DB r 216 range (bottom panel) is plotted in Fig. 9. The distribution clearly It could be debated that arguing on the existence of a population shows that the co-orbital region approximately goes from 19.0 to of high orbital inclination Uranian co-orbitals based solely on the 19.4 au. Outside that range in semimajor axis, most trajectories be- discovery of 2015 DB216 is an exercise of mere speculation. In order come passing orbits. However, even deep inside Uranus’ co-orbital to investigate this interesting hypothesis further, we have studied zone not all the values of the semimajor axis are equally favourable the evolution of a sample of 103 fictitious bodies with initial orbits regarding stability. Fig. 10 shows the distribution for the average similar to that of 2015 DB216. Their orbital elements have been value of the semimajor axis. The extension of the co-orbital zone is generated using uniformly distributed random numbers in order to confirmed, but those orbits with values of the osculating semimajor survey the relevant region of the orbital parameter space evenly. For axis in the range 19.1–19.2 au are far more stable. 4 each test orbit, a numerical integration for 10 yr – using the same Although the previous analysis strongly suggests that 2015 DB216 physical model and techniques applied in previous sections – has is not a statistical accident and that more temporary Uranian co- been performed. orbitals must exist at high orbital inclinations, our relatively short The average value (bottom panels) of the resonant angle, λr, and non-extensive integrations appear to leave the question of long- and its standard deviation (top panels) as a function of the initial term stability open. Can we expect that objects moving in high- values of the orbital parameters a, e,andi is plotted in Figs 6–8, inclination orbits like that of 2015 DB216 willspend10or100Myr respectively. The assumed ranges in a, e,andi are displayed;  trapped in the 1:1 mean motion resonance with Uranus? Figs 11

MNRAS 453, 1288–1296 (2015) A recurring co-orbital to Uranus 1295 Downloaded from

Figure 9. Distribution in initial semimajor axis for objects with standard ◦ ◦ Figure 12. Average value of the semimajor axis (top panel) and the eccen- deviation of the resonant angle, σλ , in the range 100 –108 (likely passing r tricity (bottom panel) as a function of the variation of the semimajor axis orbits, top panel) and outside that range (bottom panel). Out of 103 orbits over time. σ ∈ ◦, ◦ studied, 346 have λr (100 108 ).

and 12 suggest an answer in the affirmative. In order to study the http://mnras.oxfordjournals.org/ variation over time of a given orbital parameter, we have computed the absolute value of the difference between the initial and final values of the parameter and divided by the integrated time. Fig. 11 shows these drifts in a, e,andi per Myr. It is obvious that, taking into account the span of the co-orbital zone, relatively long-term stability is possible. Fig. 12 shows the average values of a and e as a function of the drift in a. From there, the most stable test orbits are found for the ranges in a and e of 19.0–19.6 au and 0.15–0.35, respec- at Northwestern University Library on February 2, 2016 tively. The range in a appears to be somewhat in conflict with the values found above, but there is an additional type of co-orbital motion that does not require libration of the resonant angle: mi- nor bodies following passing orbits with small Jacobi constants but still moving in unison with a host planet as described by Namouni (1999). This orbital regime is also known as the Kozai domain be- cause it corresponds to a Kozai resonance (Kozai 1962). Under the Kozai resonance, both eccentricity and inclination oscillate with the same frequency but out of phase; when the value of the eccentricity Figure 10. Same as Fig. 9 but for the average value of the semimajor axis. reaches its√ maximum the value of the inclination is the lowest and vice versa ( 1 − e2 cos i ∼ constant); therefore, relatively large os- cillations in e and i are still compatible with long-term stability in this case. The most stable test orbit generated in our exploratory calculations could remain virtually unchanged for time-scales well in excess of 10 Myr (see Fig. 12) and it is not a classical (librating) co-orbital but a fictitious object in the Kozai domain. The actual val- ues of the semimajor axis, 19.2 au, and eccentricity, 0.32, of 2015 DB216 place this object in the most stable region of the orbital do- main probed in Fig. 12. If such orbits represent nearly 0.3 per cent of the ones explored in this section and one actual object has al- ready been found, 2015 DB216, a significant number of less-stable high-inclination, recurring Uranian co-orbitals are likely to exist.

7 CONCLUSIONS

In this paper, we have analysed the orbital behaviour of 2015 DB216 that is the fourth known minor body to be trapped in a 1:1 mean mo- tion resonance with Uranus. Our numerical integrations show that it Figure 11. Variation of the eccentricity (top panel) and the inclination currently moves in a complex, horseshoe-like orbit when viewed in (bottom panel) over time as a function of the corresponding variation in a frame of reference corotating with Uranus. The object exhibits res- semimajor axis (see the text for details). onance angle nodding as it undergoes recurrent co-orbital episodes with Uranus. Its high orbital inclination clearly separates this

MNRAS 453, 1288–1296 (2015) 1296 C. de la Fuente Marcos and R. de la Fuente Marcos object from the other three known Uranian co-orbitals and makes it Gallardo T., 2006, Icarus, 184, 29 more stable. Its discovery circumstances also single this minor body Gallardo T., 2014, Icarus, 231, 273 out among objects currently trapped into the 1:1 commensurability Giorgini J. D. et al., 1996, BAAS, 28, 1158 with Uranus, hinting at the presence of a large number of similar Granvik M., Vaubaillon J., Jedicke R., 2012, Icarus, 218, 262 objects. If they are inherently more stable at higher inclinations, Grav T., Holman M. J., Gladman B. J., Aksnes K., 2003, Icarus, 166, 33 Henon M., 1969, A&A, 1, 223 that should have an impact on the population of Uranian irregular Holman M. J., Wisdom J., 1993, AJ, 105, 1987 moons. All but one are retrograde and their orbital inclinations are Ito T., Tanikawa K., 2002, MNRAS, 336, 483 in the range 139◦–167◦ or in prograde terms 41◦–13◦. Five irregular ◦ ◦ Jackson J., 1913, MNRAS, 74, 62 moons out of nine have inclinations in the range 139 –147 .As Ketchum J. A., Adams F. C., Bloch A. M., 2013, ApJ, 762, 71 for the mechanism responsible for the activation of the co-orbital Kortenkamp S. J., 2005, Icarus, 175, 409 states, multibody mean motion resonances trigger the transitions as Kozai Y., 1962, AJ, 67, 591 previously observed for other Uranian co-orbitals. Kwiatkowski T. et al., 2009, A&A, 495, 967 Lee M. H., Peale S. J., 2002, ApJ, 567, 596 Makino J., 1991, ApJ, 369, 200 ACKNOWLEDGEMENTS Marzari F., Tricarico P., Scholl H., 2003, A&A, 410, 725 Murray C. D., Dermott S. F., 1999, Solar System Dynamics. Cambridge Downloaded from We thank the anonymous referee for his/her constructive and helpful Univ. Press, Cambridge, p. 97 report, and S. J. Aarseth for providing the code used in this research. Namouni F., 1999, Icarus, 137, 293 In preparation of this paper, we made use of the NASA Astrophysics Nesvorny´ D., Dones L., 2002, Icarus, 160, 271 Data System, the astro-ph e-print server, and the MPC data server. Nesvorny´ D., Alvarellos J. L. A., Dones L., Levison H. F., 2003, AJ, 126, 398

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MNRAS 453, 1288–1296 (2015)